Abstract: The knowledge of the nuclear symmetry energy of hot neutron-rich matter isimportant for understanding the dynamical evolution of massive stars and thesupernova explosion mechanisms. In particular, the electron capture rate onnuclei and-or free protons in presupernova explosions is especially sensitiveto the symmetry energy at finite temperature. In view of the above, in thepresent work we calculate the symmetry energy as a function of the temperaturefor various values of the baryon density, by applying a momentum-dependenteffective interaction. In addition to a previous work, the thermal effects arestudied separately both in the kinetic part and the interaction part of thesymmetry energy. We focus also on the calculations of the mean field potential,employed extensively in heavy ion reaction research, both for nuclear and pureneutron matter. The proton fraction and the electron chemical potential, whichare crucial quantities for representing the thermal evolution of supernova andneutron stars, are calculated for various values of the temperature. Finally,we construct a temperature dependent equation of state of $\beta$-stablenuclear matter, the basic ingredient for the evaluation of the neutron starproperties.